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   <h4 class="subsectionHead"><span class="titlemark">2.2   </span> <a 
 id="x7-90002.2"></a>Beam Elements</h4>
<!--l. 258--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.2.1   </span> <a 
 id="x7-100002.2.1"></a>Beam2d element</h5>
<!--l. 259--><p class="noindent" >Beam element for 2D analysis, based on Timoshenko hypothesis. Structure should be defined in x,z plane. The internal
condensation of arbitrary DOF is supported and is performed in local coordinate system. On output,
the local end displacement and local end forces are printed. The element features are summarized in
Table&#x00A0;<a 
href="#x7-100024">4<!--tex4ht:ref: beam2dsummary --></a>.
<!--l. 265--><p class="indent" >   <hr class="figure"><div class="figure" 
>
                                                                                           
                                                                                           
<a 
 id="x7-100012"></a>
                                                                                           
                                                                                           
<!--l. 1--><p class="noindent" ><img 
src="elementlibmanual-figure1.png" alt="PIC"  
>
<br /> <div class="caption" 
><span class="id">Figure&#x00A0;2: </span><span  
class="content">Beam2d element. Definition of local c.s.(a) and definition of local end forces and local element dofs
(b).</span></div><!--tex4ht:label?: x7-100012 -->
                                                                                           
                                                                                           
<!--l. 273--><p class="indent" >   </div><hr class="endfigure">
<div class="center" 
>
<!--l. 275--><p class="noindent" >
<div class="table">
<!--l. 275--><p class="noindent" ><a 
 id="x7-100024"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-5" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-5-1g"><col 
id="TBL-5-1"></colgroup><colgroup id="TBL-5-2g"><col 
id="TBL-5-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-5-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-5-1-2"  
class="td11"><!--l. 275--><p class="noindent" ><span 
class="cmbx-10">beam2d</span>                                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-5-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-5-2-2"  
class="td11"><!--l. 275--><p class="noindent" >2D beam element                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-5-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-5-3-2"  
class="td11"><!--l. 275--><p class="noindent" >[<span 
class="cmtt-10">dofstocondense</span>&#x00A0;#<span 
class="cmr-5">(ia)</span>]                                              </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-5-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-5-4-2"  
class="td11"><!--l. 280--><p class="noindent" ><span 
class="cmtt-10">dofstocondense</span>: allows to specify local element dofs that
will be condensed. The numbering of local element dofs
is shown in fig.&#x00A0;<a 
href="#x7-100012">2<!--tex4ht:ref: beam2dfig --></a>. The size of this array should be equal
to  number  of  local  element  dofs  (6)  and  nonzero  value
indicates the corresponding dof will be condensed.           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-5-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-5-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-5-5-2"  
class="td11"><!--l. 283--><p class="noindent" >Three  dofs  (u-displacement,  w-displacement,  y-rotation)
are required in each node.                                           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-5-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-6-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-5-6-2"  
class="td11"><!--l. 286--><p class="noindent" >Cubic approximations of lateral displacement and rotation
are used. For longitudinal displacement the linear one is
assumed.                                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-5-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-7-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-5-7-2"  
class="td11"><!--l. 287--><p class="noindent" >Exact.                                                                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-5-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-8-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-5-8-2"  
class="td11"><!--l. 288--><p class="noindent" >Full  dynamic  analysis  support.  Linear  stability  analysis
support.                                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-5-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-9-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-5-9-2"  
class="td11"><!--l. 289--><p class="noindent" >Area,inertia  moment  along  y-axis  (<span 
class="cmtt-10">iy </span>parameter)  and
equivalent shear area (<span 
class="cmtt-10">shearareaz </span>parameter) should be
specified.                                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-5-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-10-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-5-10-2"  
class="td11"><!--l. 297--><p class="noindent" >Constant  and  linear  edge  loads  are  supported,  shear
influence  is  taken  into  account.  Edge  number  should
be  equal  to  1.  Temperature  load  is  supported,  the
first  coefficient  of  temperature  load  represent  mid-plane
temperature  change,  the  second  one  represent  difference
between  temperature  change  of  local  z+  and  local  z-
surfaces of beam (in local coordinate system). Temperature
load require that the &#8220;thick&#8221; property of cross section model
is defined.                                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-5-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-11-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-5-11-2"  
class="td11"><!--l. 298--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-5-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-12-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-5-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-5-13-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;4: </span><span  
class="content">beam2d element summary</span></div><!--tex4ht:label?: x7-100024 -->
</div><hr class="endfloat" />
</div>
</div>
   <h5 class="subsubsectionHead"><span class="titlemark">2.2.2   </span> <a 
 id="x7-110002.2.2"></a>Beam3d element</h5>
<!--l. 302--><p class="noindent" >Beam element for 3D <span 
class="cmbx-10">linear </span>analysis, based on Timoshenko hypothesis. The internal condensation of arbitrary DOF is
supported and is performed in local coordinate system. On output, the local end-displacement and local end-forces are
printed. Requires the local coordinate system to be chosen according to main central axes of inertia. Local element
coordinate system is determined by the following rules:
     <ol  class="enumerate1" >
     <li 
  class="enumerate" id="x7-11002x1">let first element node has following coordinates (<span 
class="cmmi-10">x</span><sub><span 
class="cmmi-7">i</span></sub><span 
class="cmmi-10">,y</span><sub><span 
class="cmmi-7">i</span></sub><span 
class="cmmi-10">,z</span><sub><span 
class="cmmi-7">i</span></sub>) and the second one (<span 
class="cmmi-10">x</span><sub><span 
class="cmmi-7">j</span></sub><span 
class="cmmi-10">,y</span><sub><span 
class="cmmi-7">j</span></sub><span 
class="cmmi-10">,z</span><sub><span 
class="cmmi-7">j</span></sub>),
     </li>
     <li 
  class="enumerate" id="x7-11004x2">direction vector of local x-axis is then <span 
class="cmbx-10">a</span><sub><span 
class="cmr-7">1</span></sub> = (<span 
class="cmmi-10">x</span><sub><span 
class="cmmi-7">j</span></sub> <span 
class="cmsy-10">&minus; </span><span 
class="cmmi-10">x</span><sub><span 
class="cmmi-7">i</span></sub><span 
class="cmmi-10">,y</span><sub><span 
class="cmmi-7">j</span></sub> <span 
class="cmsy-10">&minus; </span><span 
class="cmmi-10">y</span><sub><span 
class="cmmi-7">i</span></sub><span 
class="cmmi-10">,z</span><sub><span 
class="cmmi-7">j</span></sub> <span 
class="cmsy-10">&minus; </span><span 
class="cmmi-10">z</span><sub><span 
class="cmmi-7">i</span></sub>),
     </li>
     <li 
  class="enumerate" id="x7-11006x3">local y-axis direction vector lies in plane defined by local x-axis direction vector (<span 
class="cmbx-10">a</span><sub><span 
class="cmr-7">1</span></sub>) and given point
     (k-node with coordinates (<span 
class="cmmi-10">x</span><sub><span 
class="cmmi-7">k</span></sub><span 
class="cmmi-10">,y</span><sub><span 
class="cmmi-7">k</span></sub><span 
class="cmmi-10">,z</span><sub><span 
class="cmmi-7">k</span></sub>)) - so called reference node,
     </li>
     <li 
  class="enumerate" id="x7-11008x4">local  z-axis  is  then  determined  as  vector  product  of  local  x-axis  direction  vector  (<span 
class="cmbx-10">a</span><sub><span 
class="cmr-7">1</span></sub>)  by  vector
     (<span 
class="cmmi-10">x</span><sub><span 
class="cmmi-7">k</span></sub> <span 
class="cmsy-10">&minus; </span><span 
class="cmmi-10">x</span><sub><span 
class="cmmi-7">i</span></sub><span 
class="cmmi-10">,y</span><sub><span 
class="cmmi-7">k</span></sub> <span 
class="cmsy-10">&minus; </span><span 
class="cmmi-10">y</span><sub><span 
class="cmmi-7">i</span></sub><span 
class="cmmi-10">,z</span><sub><span 
class="cmmi-7">k</span></sub> <span 
class="cmsy-10">&minus; </span><span 
class="cmmi-10">z</span><sub><span 
class="cmmi-7">i</span></sub>),
                                                                                           
                                                                                           
     </li>
     <li 
  class="enumerate" id="x7-11010x5">local y-axis is then determined as vector product of local z-axis direction vector by local x-axis direction
     vector.</li></ol>
<!--l. 320--><p class="noindent" >The element features are summarized in Table&#x00A0;<a 
href="#x7-110125">5<!--tex4ht:ref: beam3dsummary --></a>. <hr class="figure"><div class="figure" 
><a 
 id="x7-110113"></a>  <img 
src="elementlibmanual-figure2.png" alt="PIC"  
>
<br /> <div class="caption" 
><span class="id">Figure&#x00A0;3: </span><span  
class="content">Beam3d element. Definition of local c.s., local end forces and local element dofs numbering.</span></div><!--tex4ht:label?: x7-110113 -->
<!--l. 329--><p class="indent" >   </div><hr class="endfigure">
<div class="center" 
>
<!--l. 332--><p class="noindent" >
<div class="table">
<!--l. 332--><p class="noindent" ><a 
 id="x7-110125"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-6" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-6-1g"><col 
id="TBL-6-1"></colgroup><colgroup id="TBL-6-2g"><col 
id="TBL-6-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-6-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-6-1-2"  
class="td11"><!--l. 332--><p class="noindent" ><span 
class="cmbx-10">beam3d</span>                                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-6-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-6-2-2"  
class="td11"><!--l. 332--><p class="noindent" >3D beam element                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-6-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-6-3-2"  
class="td11"><!--l. 332--><p class="noindent" ><span 
class="cmtt-10">refnode</span>&#x00A0;#<span 
class="cmr-5">(in) </span>[<span 
class="cmtt-10">dofstocondense</span>&#x00A0;#<span 
class="cmr-5">(ia)</span>]                          </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-6-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-6-4-2"  
class="td11"><!--l. 333--><p class="noindent" ><span 
class="cmtt-10">refnode</span>:  sets  reference  node  to  determine  the  local
coordinate system of element.                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-6-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-5-1"  
class="td11">                </td><td  style="white-space:wrap; text-align:left;" id="TBL-6-5-2"  
class="td11"><!--l. 334--><p class="noindent" ><span 
class="cmtt-10">dofstocondense</span>: allows to specify local element dofs that
will be condensed. The numbering of local element dofs
is shown in fig.&#x00A0;<a 
href="#x7-110113">3<!--tex4ht:ref: beam3dfig --></a>. The size of this array should be equal
to number of local element dofs (12) and nonzero value
indicates the corresponding dof will be condensed.           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-6-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-6-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-6-6-2"  
class="td11"><!--l. 335--><p class="noindent" >Six  dofs  (u,v,w-displacements  and  x,y,z-rotations)  are
required in each node.                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-6-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-7-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-6-7-2"  
class="td11"><!--l. 336--><p class="noindent" >Cubic approximations of lateral displacement and rotation
(along   local   y,z   axes)   are   used.   For   longitudinal
displacement and the rotation along local x-axis (torsion)
the linear approximations are assumed.                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-6-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-8-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-6-8-2"  
class="td11"><!--l. 337--><p class="noindent" >Exact.                                                                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-6-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-9-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-6-9-2"  
class="td11"><!--l. 338--><p class="noindent" >Full  dynamic  analysis  support.  Linear  stability  analysis
support.                                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-6-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-10-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-6-10-2"  
class="td11"><!--l. 339--><p class="noindent" >Area,  inertia  moment  along  y  and  z  axis  (<span 
class="cmtt-10">iy  </span>and
<span 
class="cmtt-10">iz </span>parameters),  torsion  inertia  moment  (<span 
class="cmtt-10">ik </span>parameter)
and  either  cross  section  area  shear  correction  factor
(<span 
class="cmtt-10">beamshearcoeff </span>parameter)  or  equivalent  shear  areas
(<span 
class="cmtt-10">shearareay </span>and  <span 
class="cmtt-10">shearareaz </span>parameters)  are  required.
These cross section properties are assumed to be defined in
local coordinate system of element.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-6-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-11-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-6-11-2"  
class="td11"><!--l. 340--><p class="noindent" >Constant and linear edge loads are supported. Edge number
should be equal to 1. Temperature load is supported, the
first  coefficient  of  temperature  load  represent  mid-plane
temperature  change,  the  second  one  represent  difference
between  temperature  change  of  local  z+  surface  and
local  z-  surface  surface  of  beam  and  the  third  one
represent difference between temperature change of local
y+  surface  and  local  y-  surface  of  beam.  Requires  the
&#8220;thick&#8221; (measured in direction of local z axis) and &#8220;width&#8221;
(measured in direction of local y axis) cross section model
properties to be defined.                                             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-6-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-6-12-2"  
class="td11"><!--l. 341--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-6-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-13-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-6-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-6-14-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;5: </span><span  
class="content">beam3d element summary</span></div><!--tex4ht:label?: x7-110125 -->
</div><hr class="endfloat" />
</div>
</div>
                                                                                           
                                                                                           
                                                                                           
                                                                                           
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